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Study on the arithmetic discontinuous groups

算术不连续群的研究

基本信息

批准号：
14540008
负责人：
TAKEUCHI Kisao
金额：
$ 2.18万
依托单位：
Saitama University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2004
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14540008/
关键词：
Number theory Discontinuous group Algebraic number field Discriminant

项目摘要

One of the aims of this project is to determine all arithmetic Fuchsian groups Γ with signature (O;e_1,e_2,e_3,e_4) explicitely.(1)Let K be a totally real algbraic number field K of dgree n. Let A be a quaternion algebra over K such that A【cross product】_Q R〓M_2(R) 【symmetry】H^<n-1>, where H is the Hamilton quaternion algebra. Let O is an order of A. Let Γ^1(A,O) be a Fuchsian group derived from unit group O^1 of O of norm 1. If a Fuchsian group Γ is commensurable with Γ^1 (A,O), Γ is called K-arithmetic Fuchsian group. If there exists a K-arithmetic Fuchsian group Γ with signature (O;e_1,e_2,e_3,e_4), then the degree [K:Q]【less than or equal】10. Moreover, the explicit upper bound D_0 of the discriminant d(K) of such fields K is given. We have determined the imprimitive field K of degree 10 with minimum discriminant d(K). (c.f.K.Takeuchi [1])(2)We have studied the case K=Q. Let A be a quaternion algebra over Q and let O(f) be an Eichler order in A with square-free level f. Let Γ^*(A,O(f)) be the normalizer ofΓ^1(A,O(f)) in SL_2(R). We show that if Γ is a Q-arithmetic Fuchsian group. Then Γ is a subgroup of Γ^*(A,O) of finite index, (c.f.K.Takeuchi [2])Consequently, we can deternime all Q-arithmetic Fuchsian groups Γ with signature (O;e_1,e_2,e_3,e_4) explicitly.

本课题的目的之一是明确地确定所有有签名的算术富氏群Γ(O；e_1，e_2，e_3，e_4).(1)设K是Dgree n的一个全实算术数域K，A是K上的一个四元数代数，使得A[叉积]_q R〓M_2(R)[对称性]H^&lt；n-1&gt；其中H是Hamilton四元数代数.设O是A的阶，Γ^1(A，O)是由O的范数为1的单位群O^1导出的Fuchsian群.如果一个Fuchsian群Γ与Γ^1(A，O)可公度，则Γ称为K-算术Fuchsian群.如果存在一个签名为(O；e_1，e_2，e_3，e_4)的K-算术富氏群Γ，则[K：q][小于等于]10。此外，还给出了这类域K的判别式d(K)的显式上界D_0。我们用最小判别式d(K)确定了10次非本原域K。(C.f.K.Takeuchi[1])(2)我们研究了K=Q的情形，设A是Q上的四元数代数，O(F)是A中无平方水平f的Eichler阶，Γ^*(A，O(F))是Γ^1(A，O(F))在SL_2(R)中的正规化子。我们证明了如果Γ是q-算术富氏群。则Γ是有限指标Γ^*(A，O)的一个子群(参看Takeuchi[2])，因此我们可以明确地确定所有签名为(O；e_1，e_2，e_3，e_4)的q-算术富氏群Γ。